"In computer networking and computer science, bandwidth, network bandwidth, data bandwidth, or digital bandwidth is a measurement of bit-rate of available or consumed data communication resources expressed in bits per second or multiples of it," informs Wikipedia.
Bandwidth stats are only displayed in Chrome and they're probably related to the DASH support. YouTube uses adaptive streaming, so that you get the most appropriate video quality for your bandwidth, even if it fluctuates.
Google also has a video quality report page, which shows information about YouTube streaming, including some stats for each major ISP in your area. YouTube used to offer a speed test service, but this is no longer available.